No, it is not. The center of mass is is going to be located somewhere out ahead of your body towards the thing you are pushing against. It is a point in SPACE, not a point on your BODY. If you are talking about only pushing the weight of your body, then yes, it will coincide with the center of mass of your body. During a horizontal force production event (whether pushing a sled or a person, or sprinting) where is the center of mass located relative to the midfoot? (here, the point where the musculoskeletal system interacts with the stationary earth)
Yes, the force you are exerting has a horizontal component. It's horizontal component is in fact, directly proportional to the displacement of the center of mass away from directly above the midfoot. This is a consequence of the way the human body works. The musculoskeletal system exerts a force vector pointing from it's contact with the stationary ground directly at the center of mass of the system. The result of this, plus the force of gravity (pointing directly down), is horizontal translation. If the vector points in any other direction than a line pointing straight from the midfoot to the center of mass, the result is not translation, but a rotation. If you push too high on a bottom heavy object, you upend it. This works the same for the unweighted human body. It can be represented as a point at the center of mass, acted upon by gravity and a vector pointing up from the ground at the midfoot contact point. The more horizontal force is produced, the further this point is from the midfoot.2) The force vector does intersect the center of mass with a big but, but, the force vector is not coming perpendicularly from the ground. In sprinting sports you're going to have a large vertical component, but it will also have a significant horizontal component. When I sprint I come away with a stride that's over 2 meters long and I'll have a vertical oscillation of about 10-20cm.
Exerting force not perfectly perpendicular to gravity is a simple matter of body positioning. There is nothing special about the vertical direction except when using gravity as a resistance. The lifts do not train your body to exert forces perpendicular to the ground. Your body does not know the definition of perpendicular. They train them to exert force, full stop.
The attachment is not showing the vectors correctly. If the vectors from the foot pointed in the direction indicated, the resultant force would not be horizontal motion, but torque about the center of mass. This diagram implies a moment arm between the feet and the COM. No such moment arm exists in reality (at least, not appreciably. A gait physiologist might disagree.
This is the crux of your error. Strength training does not operate by "strengthening ranges of motion". This is a regrettable approximation arrived at by personal trainers trying to motivate/bully their clients. The human body does not work this way. The adaptations which strength training produces increase force production capacity, which cannot be "abbreviated." A gluteus muscle which doubles the force it produces by contraction will double it's force production capacity even if you strip it off the body and electrocute it (controlling of course for non-muscular improvements to force production.)3) The squat is going to load all muscles but it won't load them in the same positions as you will when sprinting. While everyone has a mental image of sprinters being basically at a 45 degree angle to the ground, however in reality for most of the race a sprinters torso will be vertical and the peak of force production will be from a position where the leg is fully behind the hips and almost straight (see image). This position represents a portion of the ROM that never occurs in any squat or deadlift variation. In fact, the only lift I can think of that does work that extension in any capacity is a power clean, coincidentally the one singular lift that every single sprinter and speed athlete swears by. Bolt famously hated the weight room and this was the one lift he liked.
I can see however, why you might misunderstand this, because "range of motion" is a principle enforced by many fitness voices, Starting Strength included. But this is not because if you fail to stretch your gluteii to their maximum capacity, they suddenly become weak as kittens when your hip angle dips below 60 degrees. A guy quarter squatting 600 pounds still gets quads that can quarter squat 600 pounds
There are, rather two reasons. First different muscles operate differently at different parts of the lift. The quads are more involved at the bottom of the deadlift, the traps more involved at the top. The Starting Strength lifts (i.e., "normal human movement patterns") are selected such that they take the entire musculature through limit efforts. Abbreviating the proscribed range of motion therefore definitionally leaves a muscle group inadequately stimulated. A high bar quarter squat does not represent a limit effort by the glutes, so they have no incentive to become stronger. The "weaknesses in certain ranges of motion" therefore do not represent the principle that a muscle has ranges of motion that can be strengthened independently, but rather that an abbreviated lift does not produce a full strength adaptation. The SS lifts are selected specifically to require a limit effort from every muscle involved in the force production event. A squat requires a limit effort by the entire system of the legs: therefore, the legs will become stronger in every situation they are in. Even if it is one that barely resembles the actual squat. That's why the squat is done the way it is done. The squat being a "normal human movement pattern" does not necessarily imply that all or even most lower body force production events resemble it (though because of anatomy, they DO), but rather that it represents the maximum force generation potential. Increasing that potential increases all force production the body engages in.
Trying to "load the muscles through a range of motion I use when sprinting" is akin to trying to "load the muscles with a bullet in them so I can be stronger when I'm shot." You need to change the way you think about training.
(The second reason is because abbreviating range of motion can become an undesirable training variable. If a guy takes even his quarter squat to 600 pounds, he might not get strong glutes, but he will get strong quads, because you still need to exert a lot of force to quarter squat 600 pounds. It might even make his real squat go up a little But if a guy starts out full squatting 300 pounds, and ends up quarter squatting 600 pounds...he hasn't really gotten much stronger. This is somewhat orthogonal to your concern, but is probably the most important to the actual insistence on correct form SS has)
The squat and the deadlift will do this. It has never been observed that they fail to.I guess we can be more granular: how do I best work my ability to produce maximal force from a position where my hip angle is greater than 180 degrees, particularly how might I do this in a non-explosive capacity?